Optimal. Leaf size=106 \[ -\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0574783, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {739, 780, 217, 206} \[ -\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 739
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}+\frac{\int \frac{(d+e x) \left (2 a e^2-2 c d e x\right )}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{\left (3 d e^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{\left (3 d e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.10542, size = 91, normalized size = 0.86 \[ \frac{2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x}{a c^2 \sqrt{a+c x^2}}+\frac{3 d e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 118, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+a}}}+3\,{\frac{d{e}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+a}}}+{\frac{{d}^{3}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96924, size = 531, normalized size = 5.01 \begin{align*} \left [\frac{3 \,{\left (a c d e^{2} x^{2} + a^{2} d e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{3 \,{\left (a c d e^{2} x^{2} + a^{2} d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{a c^{3} x^{2} + a^{2} c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.55404, size = 135, normalized size = 1.27 \begin{align*} -\frac{3 \, d e^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} + \frac{x{\left (\frac{x e^{3}}{c} + \frac{c^{3} d^{3} - 3 \, a c^{2} d e^{2}}{a c^{3}}\right )} - \frac{3 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}}{a c^{3}}}{\sqrt{c x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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